The concept of puncturable encryption was introduced by Green and Miers at IEEE S&P 2015. Puncturable encryption allows recipients to update their decryption keys to revoke decryption capability for selected messages without communicating with senders. From the first instantiation, puncturable encryption shows its essence for many interesting applications, such as asynchronous messaging systems, group messaging systems, public-key watermarking schemes, secure cloud emails, and many more. To eliminate the necessity of having a costly certificate verification process, Wei et al. introduced puncturable identity-based encryption at ESORICS 2019. Unfortunately, till today, there is no puncturable identity-based encryption which can withstand quantum attacks. In this paper, we aim to fill this gap in the literature by presenting the first constructions of puncturable identity-based encryption, for both selective and adaptive identity, which are secure in the standard model based on the hardness of the learning with errors problem. Design ideas of proposed constructions might prove useful to construct other lattice-based expressive puncturable encryption as well.